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Innocuous angular integal

  1. Dec 2, 2009 #1
    Hi,
    Can anyone help me to do the following integral over the unit sphere:
    [tex]\begin{equation}\nonumber
    \int d^2 n \exp\left[-(\vec{a}\cdot \hat n)(\vec{b}\cdot \hat n)-i \,\hat n\cdot \vec{c}\right]
    \end{equation}[/tex]

    where [tex]\hat n[/tex] is a unit 3-vector, and [tex]\vec{a}[/tex], [tex]\vec{b}[/tex] and [tex]\vec{c}[/tex] are arbitrary real valued constant 3-vectors of arbitrary length. To avoid any unnecessary confusion, [tex]i\equiv \sqrt{-1}[/tex]; and [tex]d^2 n=d\Omega=\sin(\theta)d\theta d\phi[/tex] is the solid angle measure.

    Any type of solution/suggestion for a solution (which is hopefully numerically stable) would be greatly appreciated. That includes any type of expansion applied to the integrand; converting it to a PDE; using complex calculus tricks, etc. etc. Of course, an analytical solution would be best.

    I lost a huge amount of time on this, but without much success. The only expansion scheme I applied successfully, works for large [tex]|\vec c|[/tex]; but making any trick work for large [tex]|\vec a| |\vec b|[/tex] (which produces a positive quadratic term in the exponent) so far has been impossible for me.

    Thanks!
     
  2. jcsd
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